By Groves M.D., Haragus M.
This text provides a rigorous lifestyles concept for small-amplitude threedimensional traveling water waves. The hydrodynamic challenge is formulated as an infinite-dimensional Hamiltonian procedure during which an arbitrary horizontal spatial course is the timelike variable. Wave motions which are periodic in a moment, assorted horizontal course are detected utilizing a centre-manifold aid strategy in which the matter is lowered to a in the community an identical Hamiltonian process with a finite variety of levels of freedom.
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Extra resources for A Bifurcation Theory for Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves
We can therefore apply the resonant version of the Lyapunov centre theorem due to Weinstein  and further developed by Moser , which states that the nonresonance condition on the eigenvalues can be replaced by the requirement that the quadratic part of the Hamiltonian is positive-definite. The following result is obtained by applying the Weinstein-Moser theorem to the reduced Hamiltonian system and to its further reduction by the symmetry S2 ; in the latter case we recover the result given by Groves [10, Theorem 5] with the nonresonance condition removed.
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